Simultaneous symplectic spectral decomposition of positive semidefinite matrices

We establish necessary and sufficient conditions on simultaneous symplectic spectral decomposition of a family of $2n \times 2n$ real positive semidefinite matrices with symplectic kernels. We also provide a precise algebraic condition on a $2n \times 2n$ real positive semidefinite matrix with symplectic kernel for orthosymplectic spectral diagonalization, which generalizes a known result for positive definite matrices.

💡 Research Summary

The paper investigates the simultaneous symplectic spectral decomposition of families of real (2n\times2n) positive semidefinite matrices that possess symplectic kernels. After recalling the basic structure of the standard symplectic space ((\mathbb R^{2n},J)) and the symplectic group (Sp(2n)), the authors review Williamson’s theorem for positive definite matrices and its extension to the semidefinite case, where a matrix (A) with a symplectic kernel can be brought by a symplectic matrix (M) to the block‑diagonal form (M^{\top} A M = D\otimes I_{2}) with a non‑negative diagonal matrix (D). The diagonal entries are called symplectic eigenvalues.

A key technical observation (Proposition 2.1) shows that for such an (A), the product (J A) is diagonalizable over (\mathbb C^{2n}) with purely imaginary eigenvalues. This property underlies the main result.

The central theorem (Theorem 3.1) gives necessary and sufficient algebraic conditions for two or more matrices to admit a common symplectic diagonalization. For two matrices (A) and (B) the conditions are: (i) both have symplectic kernels, (ii) they symplectically commute, i.e. (A J B = B J A), and (iii) the intersection (\ker A\cap\ker B) is a symplectic subspace. Under these hypotheses there exists a single symplectic matrix (M) such that (M^{\top} A M = D_{1}\otimes I_{2}) and (M^{\top} B M = D_{2}\otimes I_{2}) with non‑negative diagonal matrices (D_{1}, D_{2}). The proof proceeds by constructing the symplectic orthogonal complement (W) of (\ker A\cap\ker B), showing that (W) is invariant under both (J A) and (J B), and then using the simultaneous diagonalizability of the commuting normal operators (J A) and (J B) on the complexified space (W\oplus iW). By iteratively extracting two‑dimensional symplectic subspaces and associated basis vectors ({p_{i},q_{i}}) together with non‑negative scalars (\lambda_{i},\mu_{i}) satisfying (A p_{i}=\lambda_{i}J q_{i}), (B p_{i}=\mu_{i}J q_{i}), a full symplectic basis of (\mathbb R^{2n}) is assembled, yielding the desired matrix (M).

Part (b) of the theorem extends the result to any finite family (\mathcal F) of such matrices: simultaneous symplectic diagonalization is possible if and only if every pair in (\mathcal F) symplectically commutes and the common kernel (\bigcap_{A\in\mathcal F}\ker A) is a symplectic subspace.

Corollary 3.3 specializes to a single matrix, stating that a positive semidefinite matrix with a symplectic kernel admits an orthosymplectic diagonalization (i.e. by a matrix that is both symplectic and orthogonal) precisely when it commutes with the symplectic form, (J A = A J). This generalizes the classic Williamson result for positive definite matrices.

Theorem 3.4 addresses the positive definite case, showing that if two positive definite matrices symplectically commute and also commute in the usual sense, then all real powers of the matrices continue to symplectically commute: (A^{s} J B^{s}=B^{s} J A^{s}) for all real (s). The proof uses the fact that (A B^{-1}) is symmetric positive definite and orthosymplectically diagonalizable, which then transfers the commutation property to arbitrary powers.

Two concrete applications are discussed. First, in Gaussian quantum information theory, a pair of zero‑mean Gaussian states with covariance matrices (V_{1}) and (V_{2}) can be simultaneously brought to normal‑mode form by a common Gaussian unitary if and only if (V_{1} J V_{2}=V_{2} J V_{1}). This follows directly from Theorem 3.1. Second, for a system with a quadratic positive definite Hamiltonian expressed as a sum of matrices (M_{i}), the partition function can be written explicitly in terms of the symplectic eigenvalues (d^{